DS arithmetic

[kyuhunl]

If a>b>0, is b<2?

1) 1/a > 1/2

2) 1/a + 1/b = 1

OA:D

[GMATGuruNY]

If a>b>0, is b<2?

1) 1/a > 1/2

2) 1/a + 1/b = 1

Statement 1:
The inequality implies that a is POSITIVE, allowing us to cross-multiply:
1*2 > a*1
2 > a
Since 2>a and a>b>0, we get:
2>a>b>0
Thus, b<2.
SUFFICIENT.

Statement 2:
If b=2, we get:
1/a + 1/2 = 1
1/a = 1/2
a=2
Not possible, since the prompt requires that a>b.

If b=3, we get:
1/a + 1/3 = 1
1/a = 2/3
3 = 2a
3/2 = a
Not possible, since the prompt requires that a>b.

The cases above illustrate that b≥2 is not viable.
Thus, b<2.
SUFFICIENT.

The correct answer is D.

[fskilnik@GMATH]

If a>b>0, is b<2?

1) 1/a > 1/2

2) 1/a + 1/b = 1

$$a > b > 0\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,ab\,\, > \,\,0} \,\,\,\,\,\frac{1}{b} > \frac{1}{a}\,\,\,\,\left( * \right)$$
$$b\,\,\mathop < \limits^? \,\,2$$

$$\left( 1 \right)\,\,\,\frac{1}{a} > \frac{1}{2}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\frac{1}{b} > \frac{1}{2}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,b < 2\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.$$
$$\left( {**} \right)\,\,\,\,\,\frac{1}{b} > \frac{1}{2}\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,2b\,\, > \,\,0} \,\,\,\,\,2 > b$$

$$\left( 2 \right)\,\,\,1 = \frac{1}{a} + \frac{1}{b}\,\,\mathop < \limits^{\left( * \right)} \,\,\,\frac{2}{b}\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\,\,\,b < 2\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.$$
$$\left( {***} \right)\,\,1 < \frac{2}{b}\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,b\,\, > \,\,0} \,\,\,\,\,b < 2$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.